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To give an update, the post office changed the zip code to 40750 and sent the package via Priority Mail.
Later in the day, I got a phone call from a lady at the Advanced Resolution Center in the Marketplace (another name for Healthcare.gov). She confirmed that although the correct zip code was 40741, 40750 would work, too. She also told me that this was not the first time someone reported the zip code issue.
She said that they probably would need a P.O. Box or a Suite# for Exemption Processing moving forward, instead of a separate zip code (40741) to avoid confusion, and that she would try to resolve the issue with the post office.